# Characterizing the image of a faithful left adjoint.

If there is an adjunction between a category $C$ and $D$ such that the left adjoint $$L:C \to D$$ is faithful (but not full), can one describe the image of $L$ in terms of the co-unit of the adjunction? If it helps, in my situation the right adjoint $R$ has a further right adjoint; I am concerned with the case actually that there is faithful functor $$f:E \to F$$ and I would like characterize the image of its prolongation $$f_{!}:Set^{E^{op}} \to Set^{F^{op}}$$ (which is again faithful).

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The fixed points (i.e. $d$ such that the counit $LRd \to d$ is an isomorphism) are in the image, but in general the image is larger. Consider sets and pointed sets: $R : Set_∗ \to Set$ is the forgetful functor and $L : Set \to Set_*$ adds a formal point. The unit for a set $X$ is $X \hookrightarrow X + \*$, the counit for a pointed set $(Y,y)$ is $Y + \* \twoheadrightarrow Y$ where $\* \mapsto y$. The counit is never an isomorphism, but $L$ is essentially surjective and faithful as required. – Martin Brandenburg Jan 14 '12 at 10:49
In another example I have in mind, the co-unit is epi on the image. I wonder if this is a characterization or not. – David Carchedi Jan 14 '12 at 18:20
What do you mean by "epi on the image"? – Martin Brandenburg Jan 15 '12 at 12:11
@Martin: In your example, the left adjoint $L$ is essentially surjective, so everything is in its image (up to iso) and the co-unit is always an epimorphism. In the case I was looking at, when I know I have something in the image, the counit happened to be epi. I was wondering if the converse were true, i.e. if the co-unit of an object was epi, then it was in the essential image. – David Carchedi Jan 15 '12 at 14:54